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### 7 Toric morphisms

#### 7.1 Toric morphisms: Examples

##### 7.1-1 Morphism between toric varieties and their class groups
gap> P1 := Polytope([[0],[1]]);
<A polytope in |R^1>
gap> P2 := Polytope([[0,0],[0,1],[1,0]]);
<A polytope in |R^2>
gap> P1 := ToricVariety( P1 );
<A projective toric variety of dimension 1>
gap> P2 := ToricVariety( P2 );
<A projective toric variety of dimension 2>
gap> P1P2 := P1*P2;
<A projective toric variety of dimension 3
which is a product of 2 toric varieties>
gap> ClassGroup( P1 );
<A free left module of rank 1 on a free generator>
gap> Display(ByASmallerPresentation(ClassGroup( P1 )));
Z^(1 x 1)
gap> ClassGroup( P2 );
<A free left module of rank 1 on a free generator>
gap> Display(ByASmallerPresentation(ClassGroup( P2 )));
Z^(1 x 1)
gap> ClassGroup( P1P2 );
<A free left module of rank 2 on free generators>
gap> Display( last );
Z^(1 x 2)
gap> PicardGroup( P1P2 );
<A free left module of rank 2 on free generators>
gap> P1P2;
<A projective smooth toric variety of dimension 3
which is a product of 2 toric varieties>
gap> P2P1:=P2*P1;
<A projective toric variety of dimension 3
which is a product of 2 toric varieties>
gap> M := [[0,0,1],[1,0,0],[0,1,0]];
[ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ]
gap> M := ToricMorphism(P1P2,M,P2P1);
<A "homomorphism" of right objects>
gap> IsMorphism(M);
true
gap> ClassGroup(M);
<A homomorphism of left modules>
gap> Display(ClassGroup(M));
[ [  0,  1 ],
[  1,  0 ] ]

the map is currently represented by the above 2 x 2 matrix


#### 7.2 The GAP category

##### 7.2-1 IsToricMorphism
 ‣ IsToricMorphism( M ) ( filter )

Returns: true or false

The GAP category of toric morphisms. A toric morphism is defined by a grid homomorphism, which is compatible with the fan structure of the two varieties.

#### 7.3 Properties

##### 7.3-1 IsMorphism
 ‣ IsMorphism( morph ) ( property )

Returns: true or false

Checks if the grid morphism morph respects the fan structure.

##### 7.3-2 IsProper
 ‣ IsProper( morph ) ( property )

Returns: true or false

Checks if the defined morphism morph is proper.

#### 7.4 Attributes

##### 7.4-1 SourceObject
 ‣ SourceObject( morph ) ( attribute )

Returns: a variety

Returns the source object of the morphism morph. This attribute is a must have.

##### 7.4-2 UnderlyingGridMorphism
 ‣ UnderlyingGridMorphism( morph ) ( attribute )

Returns: a map

Returns the grid map which defines morph.

##### 7.4-3 ToricImageObject
 ‣ ToricImageObject( morph ) ( attribute )

Returns: a variety

Returns the variety which is created by the fan which is the image of the fan of the source of morph. This is not an image in the usual sense, but a toric image.

##### 7.4-4 RangeObject
 ‣ RangeObject( morph ) ( attribute )

Returns: a variety

Returns the range of the morphism morph. If no range is given (yes, this is possible), the method returns the image.

##### 7.4-5 MorphismOnWeilDivisorGroup
 ‣ MorphismOnWeilDivisorGroup( morph ) ( attribute )

Returns: a morphism

Returns the associated morphism between the divisor group of the range of morph and the divisor group of the source.

##### 7.4-6 ClassGroup
 ‣ ClassGroup( morph ) ( attribute )

Returns: a morphism

Returns the associated morphism between the class groups of source and range of the morphism morph

##### 7.4-7 MorphismOnCartierDivisorGroup
 ‣ MorphismOnCartierDivisorGroup( morph ) ( attribute )

Returns: a morphism

Returns the associated morphism between the Cartier divisor groups of source and range of the morphism morph

##### 7.4-8 PicardGroup
 ‣ PicardGroup( morph ) ( attribute )

Returns: a morphism

Returns the associated morphism between the Picard groups of source and range of the morphism morph

##### 7.4-9 Source
 ‣ Source( morph ) ( attribute )

Returns: a variety

Return the source of the toric morphism morph.

##### 7.4-10 Range
 ‣ Range( morph ) ( attribute )

Returns: a variety

Returns the range of the toric morphism morph if specified.

##### 7.4-11 MorphismOnIthFactor
 ‣ MorphismOnIthFactor( morph ) ( attribute )

#### 7.5 Methods

##### 7.5-1 UnderlyingListList
 ‣ UnderlyingListList( morph ) ( operation )

Returns: a list

Returns a list of list which represents the grid homomorphism.

#### 7.6 Constructors

##### 7.6-1 ToricMorphism
 ‣ ToricMorphism( vari, lis ) ( operation )

Returns: a morphism

Returns the toric morphism with source vari which is represented by the matrix lis. The range is set to the image.

##### 7.6-2 ToricMorphism
 ‣ ToricMorphism( vari, lis, vari2 ) ( operation )

Returns: a morphism

Returns the toric morphism with source vari and range vari2 which is represented by the matrix lis.

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